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Mirrors > Home > MPE Home > Th. List > fvopab4ndm | Structured version Visualization version GIF version |
Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.) |
Ref | Expression |
---|---|
fvopab4ndm.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
Ref | Expression |
---|---|
fvopab4ndm | ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvopab4ndm.1 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | dmeqi 5810 | . . . 4 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
3 | dmopabss 5824 | . . . 4 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
4 | 2, 3 | eqsstri 3959 | . . 3 ⊢ dom 𝐹 ⊆ 𝐴 |
5 | 4 | sseli 3921 | . 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴) |
6 | ndmfv 6798 | . 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅) | |
7 | 5, 6 | nsyl5 159 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ∅c0 4261 {copab 5140 dom cdm 5588 ‘cfv 6430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-dm 5598 df-iota 6388 df-fv 6438 |
This theorem is referenced by: fvmptndm 6899 |
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